Now we turn back to the case of the basic wave patterns with vacuum
states. As pointed out by Liu-Smoller [1]}, among the two
nonlinear waves, i.e., shock and rarefaction waves, to the
one-dimensional compressible isentropic Euler equations (REF ),
only the rarefaction wave can be connected to vacuum. However, to
our knowledge, so far there is no any results on the zero
dissipation limit of the system (REF ) in the case when the
Euler system (REF ) contain the rarefaction wave connected to
the vacuum. In this paper, we investigate this fundamental problem
and want to obtain the decay rate with respect to the viscosity
\(\epsilon \) . Remark that Perepelitsa [2]} consider the time-asymptotic
stability of solutions of 1-d compressible Navier-Stokes equations
(REF ) toward rarefaction waves connected to vacuum in
Lagrangian coordinate and Jiu-Wang-Xin [3]} study the large
time asymptotic behavior
toward rarefaction waves for solutions to
the 1-dimensional compressible
Navier-Stokes equations (REF ) with density-dependent viscosity
for general initial data whose far fields are connected by a
rarefaction wave to the corresponding Euler equations with one end
state being vacuum.
